Introduction

Quantum control is an essential ingredient of modern quantum science and technology. The quest of manipulating localization and transport dynamics, in particular, has significantly enriched our understanding of quantum matter1,2,3, with ample implications for quantum device design4,5,6,7,8,9,10. In the region of cold atoms, the development of optical lattices greatly facilitates the exploration of topological transport11,12,13 in real-space lattice potentials. Compared to conventional optical lattices, synthetic momentum lattices feature flexible lattice design, reduced experimental complexity, and highly malleable parameters such as the site-resolved hopping patterns, quasiperiodicity and disorder, as well as variant forms of synthetic gauge fields. The momentum lattice is therefore an important platform for simulating and exploring transport dynamics in a wealth of physical contexts14,15. Over the past decade, momentum-lattice engineering has helped to reveal rich quantum dynamic phenomena regarding topological matter16,17, non-Hermitian physics18,19, correlated dynamics in frustrated geometry20, as well as localization and mobility edges21,22,23. A prime achievement here is the experimental demonstration of the inverse Anderson localization24,25,26,27,28, where disorder counterintuitively enhances delocalization in flat-band lattices via Aharonov-Bohm (AB) caging mechanisms29,30,31,32. Recently implemented in photonics33,34 and cold atoms in twisted optical lattices35, systems with flat bands exhibit striking localization and transport properties due to their dispersionless energy spectra. Despite these advances, a critical question remains–whether the transport properties of flat-band lattices can be dynamically tuned or switched between localized and delocalized regimes–a challenge central to quantum manipulation and simulation.

In this work, we experimentally demonstrate chiral transport in a synthetic rhombic chain in the momentum space of ultracold 87Rb atoms. Adopting the state-of-the-art momentum-lattice engineering technique, we are able to switch and tune the synthetic flux threaded through each plaquette on demand and probe the dynamics of the condensate along the synthetic lattice. We observe local breathing modes and biased oscillations in a binary staggered-flux configuration, both characteristic of the flat-band localization under the staggered flux. We then engineer a state-dependent chiral transport by introducing Floquet modulations to the flux. While the micromotion of the Floquet dynamics involves the occupation of multiple sublattice sites and is complicated in general, the observed chiral transport is consistent with the quantized winding of the Floquet Bloch bands of a coarse-grained effective Hamiltonian. While the topological robustness and fast time scale of the quantized transport can be useful for quantum device design, our setup further paves the way for the exploration of the rich dynamics under the impact of the flat-band localization, disorder, Floquet driving, and long-range interactions typical of the momentum lattice. To the best of our knowledge, the observed topological transport in a system with flat-band localization is the first of its kind and absent in all existing studies.

Results

Implementing the staggered-flux lattice

In our experiment, the flat bands are induced by implementing synthetic fluxes in a rhombic chain. The manipulation of synthetic flux in ultracold atomic systems is a powerful tool for manipulating band structures and dynamics, as well as for simulating and studying topological phases. For instance, synthetic flux plays key roles in generating flat bands in the AB cage26 and the Creutz lattice36, where particles exhibit highly localized states due to interference. Synthetic flux is also crucial for the simulation of the Hofstadter model37, which features fractal spectra and topological bands. The interplay of different forms of synthetic flux, lattice geometry and interaction can also give rise to dynamic phases and transitions20,38,39,40. In open systems, the combination of synthetic flux and local loss leads to directional dynamics signaling important non-Hermitian effects19.

Here we consider a binary-flux-ladder (BFL) configuration with recurring synthetic fluxes ϕ1 and ϕ2, as illustrated in Fig. 1a. Each unit cell contains 6 sublattice sites, respectively labeled {A2j, B2j, C2j, A2j+1, B2j+1, C2j+1} in the j-th unit cell. Under the tight-binding approximation41,42, the system is described by the effective Hamiltonian

$$\begin{array}{rcl}{H}_{{{\rm{eff}}}}&=&\mathop{\sum}\limits_{j}\left[-J\left({e}^{i{\theta }_{2j-1,3}}{\hat{a}}_{2j}^{{\dagger} }{\hat{b}}_{2j-1}+{e}^{-i{\theta }_{2j,4}}{\hat{a}}_{2j}^{{\dagger} }{\hat{b}}_{2j}\right.\right.\\ &&+{e}^{-i{\theta }_{2j-1,2}}{\hat{a}}_{2j}^{{\dagger} }{\hat{c}}_{2j-1}+{e}^{i{\theta }_{2j,1}}{\hat{a}}_{2j}^{{\dagger} }{\hat{c}}_{2j}+{e}^{i{\theta }_{2j,3}}{\hat{a}}_{2j+1}^{{\dagger} }{\hat{b}}_{2j}\\ &&+{e}^{-i{\theta }_{2j+1,4}}{\hat{a}}_{2j+1}^{{\dagger} }{\hat{b}}_{2j+1}+{e}^{-i{\theta }_{2j,2}}{\hat{a}}_{2j+1}^{{\dagger} }{\hat{c}}_{2j}+\\ &&\left.\left.{e}^{i{\theta }_{2j+1,1}}{\hat{a}}_{2j+1}^{{\dagger} }{\hat{c}}_{2j+1}+{{\rm{H}}}.{{\rm{c}}}.\right)\right]\end{array}$$
(1)

where J is the nearest-neighbor hopping rate, \({\hat{a}}_{2j/2j+1}^{{\dagger} }\), \({\hat{b}}_{2j/2j+1}^{{\dagger} }\) and \({\hat{c}}_{2j/2j+1}^{{\dagger} }\) (\({\hat{a}}_{2j/2j+1}\), \({\hat{b}}_{2j/2j+1}\) and \({\hat{c}}_{2j/2j+1}\)) are the creation (annihilation) operators for atoms on the sublattice sites of the j-th unit cell, respectively. Here {An, Bn, Cn, An+1} encircle the n-th plaquette, consistent with Fig. 1a. Since the synthetic flux of the n-th plaquette ϕn is given by \({\phi }_{n}(t)=\mathop{\sum }_{m = 1}^{4}{\theta }_{n,m}(t)\), we henceforth adopt the gauge convention: θn,2 = ϕ(t), and θn,m = 0 otherwise.

Fig. 1: Schematic of the rhombic flux-staggered chain and the breathing mode.
Fig. 1: Schematic of the rhombic flux-staggered chain and the breathing mode.
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a Schematics of a rhombic flux-staggered chain with flexibly programmable artificial gauge fields. The subfigure shows the binary flux-staggered ladder configuration with a static synthetic flux setting. The complex hopping coefficients between the nearest-neighbor sites are denoted as \(J{e}^{i{\theta }_{m}}(m=1,2,3,4)\). b–d The energy dispersion E(k) of the binary-flux lattice chain with ϕ2 = π. All the bands display complete flat-band features no matter what ϕ1 is. b When ϕ1 ≠ ± π, the binary-flux lattice has six flat bands, labeled as E±2, E±1, E0 respectively. The central band with E0 is two-fold degenerate. Here we choose ϕ1 = 0, under which the splitting of the upper and lower bands is the largest. d The flat-band splitting with the variation of ϕ1. e The evolution dynamics for an initial state \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle =\frac{1}{\sqrt{2}}({\hat{a}}_{n}^{{\dagger} }+{\hat{a}}_{n+1}^{{\dagger} })\left\vert 0\right\rangle\) with the binary-flux parameter {ϕ1 = 0, ϕ2 = π}. The color map represents the measured atom density ρ/ρt, normalized by the total density ρt at each discrete time slice. f Measuring a damped oscillation of the total population of sites A0 and A1 around 0.9 ms. The red solid lines represent the fitting of experimental data, and the pale red line is the numerical results calculated using Heff in Eq. (1)

The band dispersions of Hamiltonian (1) are E0 = 0, \({E}_{\pm 1}=\pm J\sqrt{4-\sqrt{2F({\phi }_{1},{\phi }_{2})}}\), and \({E}_{\pm 2}=\pm J\sqrt{4+\sqrt{2F({\phi }_{1},{\phi }_{2})}}\). Here the flux-dependent factor \(F({\phi }_{1},{\phi }_{2})=2+\cos ({\phi }_{1})+\cos ({\phi }_{2})+4\cos (2k)\cos ({\phi }_{1}/2)\cos ({\phi }_{2}/2)\), k (− π≤k < π) is the Bloch wave number. For convenience, we have set the lattice constant to unity. Notably, when ϕ2 = π, \(F({\phi }_{1},{\phi }_{2})=2{\cos }^{2}({\phi }_{1}/2)\) for arbitrary ϕ1. All the bands are then flat regardless of ϕ1 [see Fig. 1b–d]. When ϕ1 ≠ ± π, the binary-flux lattice has six flat bands with the central band being two-fold degenerate. As shown in Fig. 1b, we label these flat-bands as E±2, E±1, E0 respectively. Through changing the values of ϕ1, we can adjust the splitting between E±2 and E±1 as shown in Fig. 1d. This gives rise to intricate localized states and opens up room for engineering the transport dynamics, as we show below.

Experimentally, we implement the rhombic chain in Fig. 1a along a Raman-coupled momentum lattice in a 87Rb Bose-Einstein condensate (BEC), which contains ~ 2 × 105 atoms43,44,45. As illustrated in Fig. 1a, sublattice sites {An, Bn, Cn} are encoded in the momentum states of atomic hyperfine levels (see Methods): An and Bn correspond to different momentum states of \(\left\vert F=1,{m}_{F}=0\right\rangle\), and Cn is encoded in those of \(\left\vert F=2,{m}_{F}=0\right\rangle\), respectively. Hopping between adjacent momentum-lattice sites is implemented by two-photon Raman or Bragg processes26,46, with a fixed hopping amplitude J/h = 0.95(5) kHz. The flux ϕn is meticulously controlled by locking the relative phases of the coupling lasers. The spatial (and temporal later) inhomogeneity of the flux pattern requires a much-improved stability in phase locking, compared to previous studies with a homogeneous flux26. More details about implementing the Raman lattice are illustrated in the “Methods” section and Supplementary materials46.

Breathing mode and biased oscillation

An outstanding dynamic signature of the flat-band configuration is the localized breathing mode under the AB-caging mechanism26,47,48,49. The conventional AB caging occurs for ϕ1 = ϕ2 = π but persists under the BFL configurations with only ϕ2 = π. We experimentally confirm this by focusing on the case with ϕ1 = 0 and ϕ2 = π, which has the largest gap between the low- and high-lying bands E±2. The corresponding localized eigenstates of the two flat bands are \(\left\vert {\psi }_{n,\pm 2}\right\rangle =\frac{1}{2}[({\hat{a}}_{n}^{{\dagger} }+{\hat{a}}_{n+1}^{{\dagger} })\pm \frac{\sqrt{6}}{6}(2{\hat{b}}_{n}^{{\dagger} }+2{\hat{c}}_{n}^{{\dagger} }-{\hat{b}}_{n-1}^{{\dagger} }+{\hat{c}}_{n-1}^{{\dagger} }+{\hat{b}}_{n+1}^{{\dagger} }+{\hat{c}}_{n+1}^{{\dagger} })]\left\vert 0\right\rangle\). As such, an initial state \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle =\frac{1}{\sqrt{2}}({\hat{a}}_{n}^{{\dagger} }+{\hat{a}}_{n+1}^{{\dagger} })\left\vert 0\right\rangle\) can be expressed as an equal-weight superposition of \(\left\vert {\psi }_{n,\pm 2}\right\rangle\) and the subsequent breathing mode should acquire a frequency proportional to the energy gap between the bands. Note that similar phenomena have been observed in photonic systems50,51.

We prepare the initial state \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle =\frac{1}{\sqrt{2}}({\hat{a}}_{0}^{{\dagger} }+{\hat{a}}_{1}^{{\dagger} })\left\vert 0\right\rangle\), by coherently splitting the initial BEC on site B0 onto sites A0 and A1, and detect the density dynamics as shown in Fig. 1e. While the ensuing dynamics is localized and centered around the initial site, the breathing-mode character is clearly visible in Fig. 1f. The observed breathing mode features a frequency ω = 2π × 4.9(2) kHz, consistent with the numerically calculated energy gap ω = 2π × 4.9J in Fig. 1d. Due to the decoherence of the Raman-coupling processes, the overall dynamics is damped, with a fitted decay lifetime of 0.40(5)ms, as illustrated in Fig. 1f. The limited lifetime of decoherence is mainly caused by the inhomogeneous interactions of trap potential and the phase noise of Raman-Bragg lasers46.

For a more general choice of ϕ1 and local initial state, the dynamics of the condensate center of mass becomes oscillatory but is biased in direction depending on the initial state. For instance, Fig. 2a, b display the density dynamics under {ϕ1 = 0, ϕ2 = π} with different initial-site excitations. While the dynamics is largely localized within the initial unit cell under the flat-band condition (enforced by ϕ2 = π), atoms initialized on-site A2j (A2j+1) move toward the right (left) at the beginning of the time evolution, and the overall oscillation is accordingly biased in direction. The biased oscillation can be interpreted as an interference effect between the upper and lower legs of the rhombic plaquette, which is fully destructive (constructive) under the π (0) flux. As a result, the initial condensate prepared on-site A2j (A2j+1) tends to delocalize on the 0-flux plaquette, moving to the right (left) at short times.

Fig. 2: Biased oscillations in the binary flux-staggered lattice.
Fig. 2: Biased oscillations in the binary flux-staggered lattice.
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a The evolution dynamics under {ϕ1 = 0, ϕ2 = π}, with the initial state \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle ={\hat{a}}_{0}^{{\dagger} }\left\vert 0\right\rangle\). b The evolution dynamics under {ϕ1 = 0, ϕ2 = π} with \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle ={\hat{a}}_{1}^{{\dagger} }\left\vert 0\right\rangle\). c The evolution dynamics under {ϕ1 = 2π/3, ϕ2 = π} with \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle ={\hat{a}}_{0}^{{\dagger} }\left\vert 0\right\rangle\). d The extracted \({{\mathcal{D}}}(t)\) with \(\left\vert {\psi }_{{{\rm{ini}}}}\right\rangle ={\hat{a}}_{0}^{{\dagger} }\left\vert 0\right\rangle\) in different BFL parameters setting. The dots are experimental data, and solid lines are numerical simulations using Heff

Further, by varying ϕ1, we can adjust the oscillated dynamics of the local oscillation, as shown in Fig. 2c. This becomes clearer in Fig. 2d, where we plot the evolution of second moment of the position operator, defined through \({{\mathcal{D}}}(t)={\left[{\sum }_{n}{(n-{n}_{0})}^{2}(\left\langle {a}_{n}^{{\dagger} }{a}_{n}\right\rangle +\left\langle {b}_{n}^{{\dagger} }{b}_{n}\right\rangle +\left\langle {c}_{n}^{{\dagger} }{c}_{n}\right\rangle )\right]}^{\frac{1}{2}}\), where n0 is the position of initial injection state, \(\left\langle {a}_{n}^{{\dagger} }{a}_{n}\right\rangle\), \(\left\langle {b}_{n}^{{\dagger} }{b}_{n}\right\rangle\) and \(\left\langle {c}_{n}^{{\dagger} }{c}_{n}\right\rangle\) are the expectation values of particle number operators, respectively. We further extract the two frequency components Fig. 2d, with ω1 = 2π × 1.4(1) kHz, ω2 = 2π × 2.6(1) kHz for the {ϕ1 = 0, ϕ2 = π} case and ω1 = 2π × 1.7(1) kHz, ω2 = 2π × 2.3(1) kHz for the {ϕ1 = 2π/3, , ϕ2 = π} case, respectively. These are consistent with the theoretical prediction of \((\left\vert {E}_{\pm 2}\right\vert \pm \left\vert {E}_{\pm 1}\right\vert )/h\)46, explaining the ϕ1 dependence of the biased oscillation.

Topological chiral transport

The local dynamics observed above can be harnessed for quantized chiral transport through the Floquet engineering. We show that a topological chiral transport can be achieved by constructing a helical Floquet channel52 which leads to a perfect spin-momentum locking through the winding of the Floquet Bloch bands. The helical Floquet channel is the Floquet counterpart of the helical edge states in 2D topological insulators with time-reversal symmetry53,54,55. These states feature directional quantum transport with intrinsic spin-dependent propagation, making them ideal candidates for quantum device design.

For this purpose, we map the central two sites A2j and A2j+1 in each unit cell to pseudospins, with the spinor field operator ψn = (ψn↑, ψn↓) as shown in Fig. 3a. Then, we apply the Floquet protocol by exchanging the neighboring flux to achieve the helical Floquet channel in our BFL configuration. Under this mapping, the chirality of the transport–the propagation directions of atoms are locked with the spin degrees of freedom, is linked to the breaking of the time-reversal symmetry defined by the combination of spin reversal and complex conjugation by preparing different initial states56.

Fig. 3: Floquet channel and topological chiral transport.
Fig. 3: Floquet channel and topological chiral transport.
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a The Floquet protocol by exchanging the adjacent synthetic flux at the spectroscopic moments. In the first half T/2 period, the system can be described by the Hamiltonian H1; In the second half T/2 period, the system is switched to the Hamiltonian H2. b The theoretical results of optimal flux setting with perfect population transfer. c The calculated optimal half Floquet period time Topt with perfect transfer corresponding to each ϕopt. d The transfer dynamics within the spinor pair with different ϕ1 settings. The subfigure (d1) shows the population transfer under the {ϕ1 = 0.2π, ϕ2 = π} BFL configuration. The last three subfigures (d2–d4) are corresponding to different optimal ϕ1 = ϕopt settings with μ=1, ν = 2, 3, 4, respectively (ϕopt,11 is an imaginary number and thus is irrelevant). e, f The experimental results of the Floquet channel with right and left chiral transport, respectively. For the reason that we mainly observe the chiral current represented by the spinor pair sites An, it is unnecessary to distinguish the micromotions of atomic density in the Bn and Cn sites. So, we have combined the display of Bn and Cn sites by indicating their total density information. g The quasi-energy spectrum of the Floquet channel protocol with Ω = 2π/T. h The extracted \({{\mathcal{D}}}(t)\) curve of experimental result for the chiral transports of the Floquet channels. To indicate the chiral feature, we add ± signs in the front of D(t) to differentiate the leftward and rightward motions. The solid lines show the numerical simulations. The error bars present the standard deviation of measurements

Specifically, assuming the wave packet is initialized on a local sublattice site A2j, a perfect population transfer between A2j and A2j+1 can be achieved under appropriate flux parameters and at a discrete set of times. Analytically, we find that, following an evolution time t, the wavefunction on A2j and A2j+1 are respectively given as \([\cos ({\varepsilon }_{1}t)+\cos ({\varepsilon }_{2}t)]/2\) and \([\cos ({\varepsilon }_{1}t)-\cos ({\varepsilon }_{2}t)]/2\), where ε1 and ε2 are the eigen energies of real-space Hamiltonian46. It follows that a perfect population transfer occurs at Topt,μν, under the conditions ε1Topt,μν = (μ + ν)π and ε2Topt,μν = μπ, where μ is a positive integer and ν is a positive odd integer. In terms of the flux parameters, the conditions for the perfect population transfer are given by {ϕ1 = ϕopt,μν, ϕ2 = π}, with

$${\phi }_{{{\rm{opt}}},\mu \nu }=2\arccos \left[\frac{2\nu (2\mu +\nu )}{{\mu }^{2}+{(\mu +\nu )}^{2}}\right],\,\,\,{{\rm{and}}}\,\,\,\mu \ge \frac{1+\sqrt{3}}{2}\nu$$
(2)

In Fig. 3b, we show the discrete values of ϕopt,μν that satisfy the conditions above. On the other hand, we also have

$${T}_{{{\rm{opt}}},\mu \nu }=\frac{\mu \pi }{J\sqrt{2(2-\cos ({\phi }_{opt,\mu \nu }/2))}}$$
(3)

which gives the time of the perfect population transfer, as shown in Fig. 3c.

Note that the above conditions are highly non-trivial, as the full dynamics generically involve the occupation of all the sublattice sites including Bn and Cn. Qualitatively, the perfect population transfer derives from the interference between different flat-band components of the initial state.

Based on the flux-dependent local dynamics, we propose to implement a spin-dependent chiral transport as follows. First, we initialize the condensate on-site An to the left of the plaquette with flux ϕopt,μν. We then let the condensate evolve along the momentum lattice, switching the synthetic fluxes ϕ1 and ϕ2 within each unit cell at integer multiples of Topt,μν. Taking ν = 1 and μ = 2 for instance, for a wavefunction initialized on-site An under the synthetic-flux configuration {ϕ1 = ϕopt,21 ≈ 0.442π, ϕ2 = π}, it propagates to the right and becomes fully localized on-site An+1 at t = Topt,21 ≈ 0.425 ms when the coupling J = h × 1.5 kHz. We then swap ϕ1 and ϕ2 so that the wavefunction continues to propagate toward the right, becoming fully localized on the central site of the next plaquette (site An+2) at t = 2Topt,21. By repeating this procedure, a persistent chiral current is realized, which is quantized at discrete time steps. By contrast, the transport is not quantized when, for instance, {ϕ1 = 0.2π, ϕ2 = π}, since the oscillatory dynamics become rather complicated, as illustrated in Fig. 3d.

For the experimental confirmation of the chiral dynamics, we excite both spin-up and spin-down components at t = 0, and observe the right- and left-going chiral transport in Fig. 3e, f, respectively. The quantized chiral transport can be understood by neglecting the complicated micromotion, and focusing on coarse-grained Floquet dynamics at discrete time steps t = 2mTopt,μν (m = 0, 1, 2. . . ). The subsequent dynamics involve only the occupation of the central two sites An and An+1 in each unit cell so that we can map them to pseudospins, with the spinor field operator ψn = (ψn↑, ψn↓). The stroboscopic Floquet dynamics in the two-dimensional spinor subspace can be described by the Floquet operator \(U={e}^{-i{H}_{2}T/2}{e}^{-i{H}_{1}T/2}\), where \({H}_{1}=\frac{\pi }{T}{\sum }_{n}{\psi }_{n}^{{\dagger} }{\sigma }_{x}{\psi }_{n}\), \({H}_{2}=-\frac{\pi }{T}{\sum }_{n}{\psi }_{n}^{{\dagger} }{S}^{+}{\psi }_{n+1}+h.c.\), and \({S}^{+}=\frac{1}{2}({\sigma }_{x}+i{\sigma }_{y})\). Here T = 2Topt,μν, and H1 and H2 respectively describe the spin-flip process under the switching of flux and the transport process during each half period. Note that coarse-grained effective Hamiltonians are widely used to deal with the stroboscopic dynamics of periodically driven systems52,57,58,59,60.

In the quasi-momentum space, we have \({U}_{k}(T,0)={e}^{-ik{\sigma }_{z}}\)52, and it follows that the Floquet Bloch Hamiltonian reads46

$${H}_{k}^{F}=\frac{i}{T}log[{U}_{k}(T,0)]=\frac{1}{T}k{\sigma }_{z}$$
(4)

The Hamiltonian \({H}_{k}^{F}\) has the intrinsic feature of spin-momentum locking in the real space, which underlies the spin-dependent chiral transport. Indeed, this unique feature can be revealed by the Floquet band structure, shown in Fig. 3g, where the quasienergy spectrum is gapless, featuring decoupled linear dispersions for the two spin species. We note that these helical Floquet channels exist at discrete time steps when the interim occupations of sites Bn and Cn are neglected. This is different from previous studies where simpler engineered SSH lattice models were used61.

In Fig. 3h, we show the measured chiral transport using the quantity \({{\mathcal{D}}}(t)\). While different spin species flow in different directions, quantized chiral transport is achieved at discrete times mT/2 (m = 0, 1, 2, . . . ), consistent with the description of the stroboscopic Floquet dynamics. The quantized transport is topologically protected by the Floquet winding numbers52,57

$${\nu }_{\sigma }=\frac{1}{2\pi i}{\oint }_{BZ}dk{{\rm{Tr}}}[{U}_{k}^{\sigma }{\partial }_{k}{U}_{k}^{\sigma {\dagger} }]$$
(5)

where \({U}_{k}^{\uparrow ,\downarrow }={e}^{\pm ik}\) are the irreducible blocks of Uk. The Floquet winding numbers reflect the state-dependent winding of quasi-energy bands as the quasi-momentum k traverses the Brillouin zone. The winding numbers can also be extracted from the measured \({{\mathcal{D}}}(t)\) in Fig. 3h46. They are given by \({\nu }_{\uparrow }={{\mathcal{D}}}(T)/2=0.83\pm 0.05\) (\({\nu }_{\downarrow }={{\mathcal{D}}}(T)/2=-0.71\pm 0.05\)) for the rightward (leftward) transport, consistent the theoretical predictions ν↑ = 1 (ν↓ = − 1).

Discussion

We report the induction of topological transport out of a flat-band lattice engineered in the momentum space of ultracold atoms. The underlying helical Floquet channel here are physically richer and technically more demanding than those in ref. 62, which features a much simpler dimer chain with no synthetic flux nor flat-band structures. Likewise, our work is also distinct in design and physical mechanism from ref. 26, where the disorder-induced delocalization does not lead to a potentially useful transport pattern.

Our approach overcomes the adiabaticity constraints inherent in conventional topological pumping13, where adiabatic variation of Hamiltonian parameters is required to achieve quantized charge transport. In contrast, our strategy enables precise modulation of the system’s topological properties without the need to slow its evolution, thereby facilitating robust topological transport even under rapidly changing conditions. This ability to engineer topological transport beyond the adiabatic regime, particularly in a flat-band that facilitates strong correlations, opens new avenues for designing quantum devices aimed at ultrafast transport63,64. Moreover, the topological robustness of quantum transport demonstrated here can also be harnessed for applications in quantum metrology65,66,67,68, such that measurements are not degraded by a broad class of environmental noise or fabrication imperfections.

For the future studies, it would be also interesting to extend the BFL configuration to more complicated structures, including the 2D topological networks69,70 and more ingenious lattice configurations for manipulating localized states71. The tuning of long-range interactions intrinsic to the momentum-lattice23,72,73,74 would also provide possibilities to study the fate of flat-band localization in the strongly correlated regime75,76. Furthermore, we can combine the accessible Feshbach resonances of 85Rb or 133Cs to tune the inter-atomic interaction strength77,78. It would be fascinating to explore the interplay of interaction, flat band, and Floquet engineering for topological transport.

Materials and Methods

Experimental implementation

A 87Rb Bose-Einstein condensate (BEC) containing approximately 2 × 105 atoms is prepared in a crossed optical dipole trap with trapping frequencies 2π × (40, 100, 115) Hz, as shown in Fig. 4a. The lattice model is mapped onto momentum-lattice states with internal hyperfine states as follows: \(\left\vert n,a\right\rangle \to \left\vert F=1,p=2n\right\rangle\), \(\left\vert n,b\right\rangle \to \left\vert F=1,p=2n+1\right\rangle\), and \(\left\vert n,c\right\rangle \to \left\vert F=2,p=2n+1\right\rangle\). Here, n denotes the index of the sublattice sites An, Bn, and Cn, while p represents the real momentum of atoms. The momentum lattice is implemented using Raman-Bragg lasers applied along the weak-trapping axis. Each laser pair consists of a single-frequency beam (solid lines) and multi-frequency beams (dashed lines), with frequencies \(\{{\omega }_{j}^{+},{\omega }_{j,p}^{-}\}\) (j = 1, 2, 3) coupling adjacent momentum states \(\left\vert p\right\rangle\) and \(\left\vert p+1\right\rangle\), where p is expressed in units of 2ℏk (k being the wave vector of the 795 nm laser) and ± indicates the propagating directions of beams. As depicted in Fig. 4b, the couplings of lattice sites in the same internal state ({An, Bn} and {Bn, An+1}) are driven by laser pair \(\{{\omega }_{1}^{+},{\omega }_{1,p}^{-}\}\)41,42, while the lattice sites in different internal states ({An, Cn} and {Cn, An+1}) are coupled by the Raman laser pairs of \(\{{\omega }_{2}^{+},{\omega }_{2,p}^{-}\}\) and \(\{{\omega }_{3}^{+},{\omega }_{3,p}^{-}\}\)26. The multi-frequency components ωj,p are finely tuned to minimize two-photon detuning.

Fig. 4: Illustration of experimental configuration and momentum-lattice encoding scheme.
Fig. 4: Illustration of experimental configuration and momentum-lattice encoding scheme.
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a The momentum lattice is constructed with three Raman-Bragg laser pairs, which are indicated with different colors and labeled by their corresponding frequencies \(\{{\omega }_{j}^{+},{\omega }_{j,p}^{-}\}\) (j = 1, 2, 3). b Configuration for the multi-frequency Raman-Bragg couplings. The solid (dashed) lines represent the single-frequency (multi-frequency) lasers. The sites An and Bn (Cn) are encoded with the momentum states of hyperfine level F = 1 (F = 2). As shown in the figure, the pairs of momentum states \(\{\left\vert n,a\right\rangle ,\left\vert n,b\right\rangle \}\) and \(\{\left\vert n,b\right\rangle ,\left\vert n+1,a\right\rangle \}\) are all coupled with laser pairs \(\{{\omega }_{1}^{+},{\omega }_{1,p}^{-}\}\), while the corresponding momentum states \(\{\left\vert n,a\right\rangle ,\left\vert n,c\right\rangle \}\), and \(\{\left\vert n,c\right\rangle ,\left\vert n+1,a\right\rangle \}\) are coupled with laser pairs \(\{{\omega }_{2}^{+},{\omega }_{2,p}^{-}\}\) and \(\{{\omega }_{3}^{+},{\omega }_{3,p}^{-}\}\), respectively. The frequencies \({\omega }_{j,p}^{-}\) are tuned to satisfy the two-photon resonant condition. The single-photon detuning is around 2.8 GHz. The hyperfine split between levels F = 1 and F = 2 of 87Rb is about 6.8 GHz

Following Raman-Bragg coupling, the atomic population distribution is measured via absorption imaging. The probe light addresses atoms in the F = 2 level, enabling site-selective detection. For imaging, both the optical dipole trap and Raman-Bragg lasers are switched off, and the atoms undergo a 20 ms time-of-flight expansion. During this period, atoms with different momenta spatially separate, allowing direct extraction of populations in each momentum state. This imaging method provides a clear visualization of the atomic distribution across lattice sites and serves as a key tool for probing the effects of Raman-Bragg coupling in the experiment.

As shown in the energy-level diagram in Fig. 4b, only the Cn sites in the lattice model are encoded in the F = 2 state. To determine the population in the individual lattice sites An, Bn and Cn, we employ a two-step measurement process. In the first step, a pumping light transfers the population of each momentum-lattice site in the F = 1 state to the F = 2 state. An absorption image of the F = 2 state then provides the combined population of both internal states for each momentum state. In the second step, we omit the pumping process and take an absorption image directly of the F = 2 state, thereby measuring only the population in the Cn sites. If resolving the population of the F = 2 state across specific lattice sites is unnecessary, the first step alone suffices, which directly yields the total population of each momentum-lattice site as illustrated in Fig. 3 of the main text. This two-step protocol offers flexibility, enabling targeted characterization of population distributions across different internal states and lattice sites according to experimental needs.